0

The Greeks were initially convinced that all numbers were rational until upon pain of contradiction were forced to accept that $\sqrt{2}$ was irrational and needed to be included in our number system for geometric reasons. All other square roots are also necessary for the same reason, and so is $\pi$. $e$ is useful for other purposes but not really necessary in order to satisfy geometric needs. So you can define and use $e$ but you really need $\sqrt{2}$ because we are committed to certain other facts in Geometry.

So my question is, if we wanted a minimal number system that could handle our needs in applied mathematics (e.g., Physics and Geometry), would we be forced to accept any numbers besides the rationals, square roots, and $\pi$?

I know that by lacking the other real numbers you have an incomplete topological space, notions of convergent sequences become harder, and so on, but that's all for mathematical ease, not for theoretical necessity. Or am I wrong about that?

Mikhail Katz
  • 42,112
  • 3
  • 66
  • 131
Addem
  • 5,656
  • Know anything about electrical engineering ? – Rene Schipperus Apr 29 '16 at 22:51
  • @Rene Schipperus Approximately no. – Addem Apr 29 '16 at 22:52
  • This is off topic because it is too hypothetical/speculative/opinion-based. Might be good in the chat, though, if not on this site then something like the xmcd forums or quora – rschwieb Apr 29 '16 at 22:53
  • 11
    I like 2. Sometimes I need 3 when the going gets tough. – Qiaochu Yuan Apr 29 '16 at 22:54
  • 2
    Do you want to have circumferences of ellipses? Then you'll need all the numbers that come up as elliptic integrals. – Gerry Myerson Apr 29 '16 at 23:06
  • 2
    Consider two problems:

    For the first, draw an equilateral triangle and let $O$ be the meeting point of its angle bisectors, and $s$ be the distance from $O$ to one of the vertices. Now bisect each of the angles $AOB$ where $A$ and $B$ are adjacent sides of the figure, and $O$ is that central point; this forms a regular polygon with twice as many sides. Repeat this "many" times. What is the area (in terms of $s$) of the union of all the polygons thus formed?

     – Mark Fischler Apr 29 '16 at 23:15
  • 2
    For the second problem, start with equal squares $ABCD$ and $BEFC$, sharing side $BC$ with $ABE$ forming a straight line. Then add a rectangle $EHIG$ to the figure, with one side extending line $ABE$ by a distance equal to $AB$, and with the "height" a line segment $EG$ along $EF$ such that the geometric mean of $AE$ and $EG$ is equal to $AD$. Keep adding rectangles on the same side: The next rectangle $HKLJ$ is such that the geometric mean of $HJ$ and $AH$ is $EG$ and so forth. What is the area (in terms of $AD$) of the union of all these squares and rectangles. – Mark Fischler Apr 29 '16 at 23:15
  • 2
    How is problem one any more acceptable than problem 2? And yet the answer to problem one involves $\pi$ while the answer to problem two involves $e$. – Mark Fischler Apr 29 '16 at 23:15
  • 8
    Addem, you should look at the computable numbers and then at the definable real numbers, for me it is obvious that those are what your question is about – reuns Apr 29 '16 at 23:50
  • 3
    This is a perfectly reasonable soft question, with a possible answer hinted on in the comment by @user1952009 therefore I vote to reopen. – Mikhail Katz May 03 '16 at 08:56
  • Addem, you should keep in mind that you can vote to reopen your own questions. – Mikhail Katz May 04 '16 at 16:49
  • Related: Are all numbers real numbers? – Dave L. Renfro May 04 '16 at 17:03

1 Answers1

1

Nice question! Admittedly, the elusive real numbers that are neither computable nor definable are perhaps of limited use in applications in physics and elsewhere (see links provided in the comments).

As you point out, formal aspects such as existence of least upper bound are harder to handle if one works with a subset of the reals.

One further extension of your question is, if we already extended the ordered number system from the naturals to the integers to the rationals to the reals, why not throw in the familiar infinitesimals to form an ordered system where analysis can be done the way Leibniz, Euler, and Cauchy did it? Such a broader number system is useful in applications, such as modeling the phenomenon of small oscillations, by giving precise meaning to the idea that the period is independent of the amplitude without engaging in paraphrases via limits; see this article in Quantum Studies: Mathematics and Foundations.

Mikhail Katz
  • 42,112
  • 3
  • 66
  • 131